Vol. 10, No. 5, 2017

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Hardy-singular boundary mass and Sobolev-critical variational problems

Nassif Ghoussoub and Frédéric Robert

Vol. 10 (2017), No. 5, 1017–1079
Abstract

We investigate the Hardy–Schrödinger operator Lγ = Δ γ|x|2 on smooth domains Ω n whose boundaries contain the singularity 0. We prove a Hopf-type result and optimal regularity for variational solutions of corresponding linear and nonlinear Dirichlet boundary value problems, including the equation Lγu = u2(s)1 |x|s , where γ < 1 4n2 , s [0,2) and 2(s) := 2(n s)(n 2) is the critical Hardy–Sobolev exponent. We also give a complete description of the profile of all positive solutions — variational or not — of the corresponding linear equation on the punctured domain. The value γ = 1 4(n2 1) turns out to be a critical threshold for the operator Lγ. When 1 4(n2 1) < γ < 1 4n2 , a notion of Hardy singular boundary mass mγ(Ω) associated to the operator Lγ can be assigned to any conformally bounded domain Ω such that 0 Ω. As a byproduct, we give a complete answer to problems of existence of extremals for Hardy–Sobolev inequalities, and consequently for those of Caffarelli, Kohn and Nirenberg. These results extend previous contributions by the authors in the case γ = 0, and by Chern and Lin for the case γ < 1 4(n 2)2 . More specifically, we show that extremals exist when 0 γ 1 4(n2 1) if the mean curvature of Ω at 0 is negative. On the other hand, if 1 4(n2 1) < γ < 1 4n2 , extremals then exist whenever the Hardy singular boundary mass mγ(Ω) of the domain is positive.

Keywords
Hardy&ndash;Schr&ouml;dinger operator, Hardy-singular boundary mass, Hardy&ndash;Sobolev inequalities, mean curvature
Mathematical Subject Classification 2010
Primary: 35J35, 35J60, 58J05, 35B44
Milestones
Received: 4 January 2016
Revised: 23 February 2017
Accepted: 3 April 2017
Published: 1 July 2017
Authors
Nassif Ghoussoub
Department of Mathematics
1984 Mathematics Road
University of British Columbia
Vancouver, BC V6T 1Z2
Canada
Frédéric Robert
Institut Élie Cartan
Université de Lorraine
BP 70239
F-54506 Vandœuvre-lès-Nancy
France